A090245 Maximum numbers of cards that would have no SET in an n-attribute version of the SET card game.

1, 2, 4, 9, 20, 45, 112

Offset: 0

Author

Hans Havermann, Jan 23 2004

Keywords

• nonn
• hard
• more
• nice

Comments

Or, largest size of an n-dimensional capset (i.e., a subset of (Z/3Z)^n that does not contain any lines {a, a+r, a+2r}). - Terence Tao, Feb 20 2009

Or, size of maximal cap in the affine geometry AG(n+1,3). - N. J. A. Sloane, Oct 25 2014

Links

• Bela Bajnok, Additive Combinatorics: A Menu of Research Problems, arXiv:1705.07444 [math.NT], May 2017.
• Robert A. Bosch, ‘Set’less Collections of SET Cards, 2000.
• D. V. Brink, The search for SET, 1997. [via WayBackMachine]
• Benjamin Lent Davis and Diane Maclagan, The Card Game SET, The Mathematical Intelligencer, Vol. 25:3 (Summer 2003), pp. 33-40.
• Ernest Davis, Review of Romera-Paredes et al., 14 Dec 2023 Nature paper about FunSearch, January 7, 2024
• Yves Edel, Caps, 2010.
• Yves Edel, Sandy Ferret, Ivan Landjev, Leo Storme, The classification of the largest caps in AG(5,3), Journal of Combinatorial Theory A, 99 (2002), 95-110.
• Jordan S. Ellenberg, Bounds for cap sets, Quomodocumque Blog, May 13 2016.
• Jordan S. Ellenberg and Dion Gijswijt, On large subsets of F_q^n with no three-term arithmetic progression, arXiv:1605.09223 [math.CO], 2016.
• Michael Follett, et al. Partitions of AG (4, 3) into Maximal Caps, Discrete Math., 337 (2014), 1-8. Preprint: arXiv:1302.4703 [math.CO].
• Dion Gijswijt, The card game SET: a mathematical challenge, 2016.
• Dion Gijswijt, The Beautiful Mathematics of the Card Game SET, STAtOR, Netherlands Society for Statistics and Operations Research (VVSOR, 2019) Vol. 20, No. 2, 10-13.
• Guardians of SET, SET Home Page.
• Pierre Jalinière, Le jeu Set, Images des Mathématiques, CNRS, 2013.
• Miriam Melnick, The Joy of SET, May 2011.
• J. Peebles, Cap Set Bounds and Matrix Multiplication, Senior Thesis poster (see also the thesis itself), Harvey Mudd College, 2013.
• Ivars Peterson, SET Math.
• Aaron Potechin, Maximal caps in AG(6, 3), Designs, Codes and Cryptography, Volume 46, Number 3, March 2008.
• Bernardino Romera-Paredes, Mohammadamin Barekatain, Alexander Novikov, et al. Mathematical discoveries from program search with large language models, Nature 625, 468-475 (2024).
• SET card game, Official web site.
• Terence Tao, Bounds for the first few density Hales-Jewett numbers, and related quantities.
• Wikipedia, Set (card game)
• M. Zabrocki, The Joy of SET, 2001.

Crossrefs

Cf. A090246, A156989.

Formula

a(n) <= A003142(n).

Asymptotically, a(n) = O(3^n/n) and a(n) > (2.21...)^n. - Terence Tao, Feb 20 2009

Asymptotically, a(n) = o(2.756^n). - David Radcliffe, May 30 2016

Extensions

a(6) from Terence Tao, Feb 20 2009

Edited by N. J. A. Sloane, Feb 21 2009

Edited by Andrey Zabolotskiy, Mar 01 2024